Chapter 2.2, Problem 53ES

Suppose that a function $f$ is differentiable at $x=0$ with $f\left(0\right)=f^{\prime }\left(0\right)=0$ , and let $y=mx,m\ne 0$ , denote any line of nonzero slope through the origin.

(a) Prove that there exists an open interval containing $0$ such that for all nonzero x in this interval $\left|f\left(x\right)\right|<\left|\frac{1}{2}mx\right|$ .

(b) Conclude from part (a) and the triangle inequality that there exists an open interval containing $0$ such that $\left|f\left(x\right)\right|<\left|f\left(x\right)-mx\right|$ for all $x$ in this interval.

(c) Explain why the result obtained in part (b) may be interpreted to mean that the tangent line to the graph of $f$ at the origin is the best linear approximation to $f$ at that point.